Difference between revisions of "Nil manifold"
From Encyclopedia of Mathematics
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A compact quotient space of a connected nilpotent Lie group (cf. [[Lie group, nilpotent|Lie group, nilpotent]]). (However, sometimes compactness is not required.) | A compact quotient space of a connected nilpotent Lie group (cf. [[Lie group, nilpotent|Lie group, nilpotent]]). (However, sometimes compactness is not required.) | ||
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Cf. also [[Nil flow|Nil flow]] and the references quoted there. | Cf. also [[Nil flow|Nil flow]] and the references quoted there. | ||
| − | An example of a nil manifold that is rather important for various applications is the following. Consider the three-dimensional Heisenberg group | + | An example of a nil manifold that is rather important for various applications is the following. Consider the three-dimensional Heisenberg group $N$ of all matrices of the form |
| − | + | $$\begin{pmatrix}1&y&z\\0&1&x\\0&0&1\end{pmatrix}$$ | |
| − | and the discrete subgroup | + | and the discrete subgroup $\Gamma$ of all such matrices with integer $x$, $y$, $z$. The corresponding quotient space $\Gamma\setminus N$ of cosets $\Gamma n$, $n\in N$, is a compact nil manifold with an invariant probability measure. It plays an important role in harmonic analysis and the theory of theta-functions. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Auslander, "Lecture notes on nil-theta functions" , Amer. Math. Soc. (1977)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Auslander, "Lecture notes on nil-theta functions" , Amer. Math. Soc. (1977)</TD></TR></table> | ||
Latest revision as of 15:48, 10 July 2014
A compact quotient space of a connected nilpotent Lie group (cf. Lie group, nilpotent). (However, sometimes compactness is not required.)
References
| [1] | A.I. Mal'tsev, "On a class of homogeneous spaces" Transl. Amer. Math. Soc. (1) , 9 (1962) pp. 276–307 Izv. Akad. Nauk SSSR Ser. Mat. , 13 : 1 (1949) pp. 9–32 |
Comments
Cf. also Nil flow and the references quoted there.
An example of a nil manifold that is rather important for various applications is the following. Consider the three-dimensional Heisenberg group $N$ of all matrices of the form
$$\begin{pmatrix}1&y&z\\0&1&x\\0&0&1\end{pmatrix}$$
and the discrete subgroup $\Gamma$ of all such matrices with integer $x$, $y$, $z$. The corresponding quotient space $\Gamma\setminus N$ of cosets $\Gamma n$, $n\in N$, is a compact nil manifold with an invariant probability measure. It plays an important role in harmonic analysis and the theory of theta-functions.
References
| [a1] | L. Auslander, "Lecture notes on nil-theta functions" , Amer. Math. Soc. (1977) |
How to Cite This Entry:
Nil manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_manifold&oldid=32406
Nil manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_manifold&oldid=32406
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article